Semiparametric deconvolution with unknown noise variance

UMR C 8628 du CNRS, Équipe de Probabilités, Statistique et Modélisation, bâtiment 425, Université Paris-Sud, 91405 Orsay Cedex, France; Catherine.Matias@math.u-psud.fr.

Abstract

This paper deals with semiparametric convolution models, where the noise sequence has a Gaussian centered distribution, with unknown variance. Non-parametric convolution models are concerned with the case of an entirely known distribution for the noise sequence, and they have been widely studied in the past decade. The main property of those models is the following one: the more regular the distribution of the noise is, the worst the rate of convergence for the estimation of the signal's density g is [5]. Nevertheless, regularity assumptions on the signal density g improve those rates of convergence [15]. In this paper, we show that when the noise (assumed to be Gaussian centered) has a variance σ² that is unknown (actually, it is always the case in practical applications), the rates of convergence for the estimation of g are seriously deteriorated, whatever its regularity is supposed to be. More precisely, the minimax risk for the pointwise estimation of g over a class of regular densities is lower bounded by a constant over log ⁿ. We construct two estimators of σ², and more particularly, an estimator which is consistent as soon as the signal has a finite first order moment. We also mention as a consequence the deterioration of the rate of convergence in the estimation of the parameters in the nonlinear errors-in-variables model.